Random Variable The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages when four software packages are randomly selected in China. What is the random variable, what are its possible values, and are its values numerical?

If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn’t select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and objects of type B is

In New Jersey’s Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probability of getting exactly four winning numbers with one ticket.

In Exercises 25–28, find the probabilities and answer the questions.

Whitus v. Georgia In the classic legal case of Whitus v. Georgia, a jury pool of 90 people was supposed to be randomly selected from a population in which 27% were minorities. Among the 90 people selected, 7 were minorities. Find the probability of getting 7 or fewer minorities if the jury pool was randomly selected. Is the result of 7 minorities significantly low? What does the result suggest about the jury selection process?

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Online Courses College students are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they take one or more online courses (based on data from Sallie Mae).

In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.

LOL In a U.S. Cellular survey of 500 smartphone users, subjects are asked if they find abbreviations (such as LOL or BFF) annoying, and each response was recorded as “yes,” “no,” or “not sure.”

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.

40% of consumers believe that cash will be obsolete in the next 20 years (based on a survey by J.P. Morgan Chase). In each of Exercises 15–20, assume that 8 consumers are randomly selected. Find the indicated probability.

Find the probability that no more than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.